Symmetric and Asymmetric Distributions: Theoretical Developments and Applications Ⅲ

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 24663

Special Issue Editors

Department of Quantitative Methods and TIDES Institute, University of Las Palmas de Gran Canaria, Campus de Tafira s/n, 35017 Las Palmas, Spain
Interests: distributions theory; Bayesian statistics; robustness; Bayesian applications in economics (actuarial, credibility, ruin theory)
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, University of Antofagasta, Antofagasta, Chile
Interests: distributions theory; bayesian statistic; classical statistic; regression
Special Issues, Collections and Topics in MDPI journals
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne, Australia
Interests: actuarial statistics; bayesian statistics; distribution theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This volume is the last number of Symmetry’s special issues devoted to symmetric and asymmetric probability distributions. This modeling includes discrete and continuous and also univariate and multivariate models. The importance of symmetry (and asymmetry) property when modeling empirical data is widely known. For example, in binary regression models with a logistic link, incorporating an asymmetry parameter can better explain the probability of the event under study and therefore improve its prediction. Many other examples might be considered, for instance, in the linear regression model, the error term could include some degree of skewness and thus moves away from the normal distribution.

 In this special issue on symmetric and asymmetric distributions, researchers working in applied statistics are welcome to submit original contributions from a wide range of applied disciplines where the importance of symmetry and asymmetry in modeling is essential. In addition, theoretical contributions on this topic are also welcome. Nowadays, with the advent of computational advances and the help of the many available statistical packages, the scope of applications has considerably grown, allowing the handling of databases that were, given their size a few decades ago, unmanageable only a few decades ago. Researchers are therefore encouraged to submit theoretical and applied contributions from different disciplines, including economics (inflation, income and wealth forecasting, stochastic frontier models, insurance, duration models, health economics, tourism, etc.), environmental sciences (including catastrophic events, climate change, etc.), biometrics, engineering (reliability, classification of satellite images, etc.), medicine (analysis of oncological diseases, cure rate models, experimental medicine, etc.) and many other areas of application. Although not limited to, this special issue aims to offer alternative methodologies to the existing modeling techniques and is open to original research and review articles, both theoretical and applied (empirical data fitting, regression, Bayesian analysis, etc.)

Prof. Dr. Emilio Gómez Déniz
Prof. Héctor W. Gómez
Dr. Enrique Calderín-Ojeda
Guest Editors

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Keywords

Applications; Bayesian; Kurtosis; Order Statistics; Regression; Simulation; Skewness

Published Papers (16 papers)

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Editorial

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4 pages, 180 KiB  
Editorial
Symmetric and Asymmetric Distributions: Theoretical Developments and Applications III
Symmetry 2022, 14(10), 2143; https://doi.org/10.3390/sym14102143 - 14 Oct 2022
Cited by 1 | Viewed by 982
Abstract
A summary of the eleven papers published in this special issue is presented here. This volume was the last in a series of special issues dealing with symmetric and non-symmetric continuous probability distributions. The works presented in this issue propose new probabilistic models [...] Read more.
A summary of the eleven papers published in this special issue is presented here. This volume was the last in a series of special issues dealing with symmetric and non-symmetric continuous probability distributions. The works presented in this issue propose new probabilistic models and extend the properties of other existing models in the statistical literature. Full article

Research

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22 pages, 1761 KiB  
Article
The Process Capability Index of Pareto Model under Progressive Type-II Censoring: Various Bayesian and Bootstrap Algorithms for Asymmetric Data
Symmetry 2023, 15(4), 879; https://doi.org/10.3390/sym15040879 - 07 Apr 2023
Viewed by 1255
Abstract
It is agreed by industry experts that manufacturing processes are evaluated using quantitative indicators of units produced from this process. For example, the Cpy process capability index is usually unknown and therefore estimated based on a sample drawn from the requested [...] Read more.
It is agreed by industry experts that manufacturing processes are evaluated using quantitative indicators of units produced from this process. For example, the Cpy process capability index is usually unknown and therefore estimated based on a sample drawn from the requested process. In this paper, Cpy process capability index estimates were generated using two iterative methods and a Bayesian method of estimation based on stepwise controlled type II data from the Pareto model. In iterative methods, besides the traditional probability-based estimation, there are other competitive methods, known as bootstrap, which are alternative methods to the common probability method, especially in small samples. In the Bayesian method, we have applied the Gibbs sampling procedure with the help of the significant sampling technique. Moreover, the approximate and highest confidence intervals for the posterior intensity of Cpy were also obtained. Massive simulation studies have been performed to evaluate the behavior of Cpy. Ultimately, application to real-life data is seen to demonstrate the proposed methodology and its applicability. Full article
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16 pages, 324 KiB  
Article
Copula Approach for Dependent Competing Risks of Generalized Half-Logistic Distributions under Accelerated Censoring Data
Symmetry 2023, 15(2), 564; https://doi.org/10.3390/sym15020564 - 20 Feb 2023
Cited by 1 | Viewed by 1018
Abstract
In medicals sciences and reliability engineering, the failure of individuals or units (I/Us) occurs due to independent causes of failure. In general, the symmetry between dependent and independent causes of failure is essential to the nature of the problem at hand. In this [...] Read more.
In medicals sciences and reliability engineering, the failure of individuals or units (I/Us) occurs due to independent causes of failure. In general, the symmetry between dependent and independent causes of failure is essential to the nature of the problem at hand. In this study, we considered the accelerated dependent competing risks model when the lifetime of I/Us was modeled using a generalized half-logistic distribution. The data were obtained with respect to constant stress accelerated life tests (ALTs) with a type-II progressive censoring scheme. The dependence structure was formulated using the copula approach (symmetric Archimedean copula). The model parameters were estimated with the maximum likelihood method; only two dependent causes of failure and bivariate Pareto copula functions were proposed. The approximate confidence intervals were constructed using both the asymptotic normality distribution of MLEs and bootstrap techniques. Additionally, an estimator of the reliability of the system under a normal stress level was constructed. The results from the estimation methods were tested by performing a Monte Carlo simulation study. Finally, an analysis of data sets from two stress levels was performed for illustrative purposes. Full article
18 pages, 1801 KiB  
Article
A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data
Symmetry 2023, 15(2), 411; https://doi.org/10.3390/sym15020411 - 03 Feb 2023
Cited by 1 | Viewed by 902
Abstract
The primary objective of this article was to introduce a new probabilistic model for the discussion and analysis of random covariates. The introduced model was derived based on the Marshall–Olkin shock model. After proposing the mathematical form of the new bivariate model, some [...] Read more.
The primary objective of this article was to introduce a new probabilistic model for the discussion and analysis of random covariates. The introduced model was derived based on the Marshall–Olkin shock model. After proposing the mathematical form of the new bivariate model, some of its distributional properties, including joint probability distribution, joint reliability distribution, joint reversed (hazard) rate distribution, marginal probability density function, conditional probability density function, moments, and distributions for both Y=max{X1,X2} and W=min{X1,X2}, were investigated. This novel model can be applied to discuss and evaluate symmetric and asymmetric data under various kinds of dispersion. Moreover, it can be used as a probability approach to analyze different shapes of hazard rates. The maximum likelihood approach was utilized for estimating the parameters of the bivariate model. A simulation study was carried out to assess the performance of the parameters, and it was noted that the maximum likelihood technique can be used to generate consistent estimators. Finally, two real datasets were analyzed to illustrate the notability of the novel bivariate distribution, and it was found that the suggested distribution provided a better fit than the competitive bivariate models. Full article
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21 pages, 989 KiB  
Article
A New More Flexible Class of Distributions on (0,1): Properties and Applications to Univariate Data and Quantile Regression
Symmetry 2023, 15(2), 267; https://doi.org/10.3390/sym15020267 - 18 Jan 2023
Cited by 1 | Viewed by 797
Abstract
In this paper, we will present a new, more flexible class of distributions with a domain in the interval (0,1), which presents heavier tails than other distributions in the same domain, such as the Beta, Kumaraswamy, and Weibull Unitary [...] Read more.
In this paper, we will present a new, more flexible class of distributions with a domain in the interval (0,1), which presents heavier tails than other distributions in the same domain, such as the Beta, Kumaraswamy, and Weibull Unitary distributions. This new distribution is obtained as a transformation of two independent random variables with a Weibull distribution, which we will call the Generalized Unitary Weibull distribution. Considering a particular case, we will obtain an alternative to the Beta, Kumaraswamy, and Weibull Unitary distributions. We will call this new distribution of two parameters the type 2 unitary Weibull distribution. The probability density function, cumulative probability distribution, survival function, hazard rate, and some important properties that will allow us to infer are provided. We will carry out a simulation study using the maximum likelihood method and we will analyze the behavior of the parameter estimates. By way of illustration, real data will be used to show the flexibility of the new distribution by comparing it with other distributions that are known in the literature. Finally, we will show a quantile regression application, where it is shown how the proposed distribution fits better than other competing distributions for this type of application. Full article
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20 pages, 1147 KiB  
Article
A More Flexible Reliability Model Based on the Gompertz Function and the Generalized Integro-Exponential Function
Symmetry 2022, 14(6), 1207; https://doi.org/10.3390/sym14061207 - 10 Jun 2022
Cited by 1 | Viewed by 1461
Abstract
This work presents a new distribution that allows modeling data from a random variable with non-negative values. The new family is defined by a stochastic representation of a scaled mixture of a random variable with a Gompertz distribution (G) and a [...] Read more.
This work presents a new distribution that allows modeling data from a random variable with non-negative values. The new family is defined by a stochastic representation of a scaled mixture of a random variable with a Gompertz distribution (G) and a random variable with a uniform distribution on the interval (0,1). The result of this gives rise to a new random variable with a Slash Gompertz (SG) distribution that is more flexible than the Gompertz distribution, that is, it better models atypical data, presenting tails heavier than the Gompertz distribution. The density and some general properties of the resulting family are studied, including its moments and kurtosis coefficient. The inference of the parameters is carried out using the method of moments and maximum likelihood. Finally, illustrations of particular cases of this family are shown, adjusting in this case two sets of real data and estimating the parameters by maximum likelihood, where it is verified that this new family of distributions fits the reliability function better than the distributions of Gompertz (G), Slash Birnbaum Saunders (SBS), Slash Weibull (SW), and Gompertz-Verhults (GV). Full article
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9 pages, 666 KiB  
Article
Some Results on the Truncated Multivariate Skew-Normal Distribution
Symmetry 2022, 14(5), 970; https://doi.org/10.3390/sym14050970 - 09 May 2022
Cited by 4 | Viewed by 1387
Abstract
The multivariate skew-normal distribution is useful for modeling departures from normality in data through parameters controlling skewness. Recently, several extensions of this distribution have been proposed in the statistical literature, among which the truncated multivariate skew-normal distribution is the foremost. Truncated distributions appear [...] Read more.
The multivariate skew-normal distribution is useful for modeling departures from normality in data through parameters controlling skewness. Recently, several extensions of this distribution have been proposed in the statistical literature, among which the truncated multivariate skew-normal distribution is the foremost. Truncated distributions appear frequently in various theoretical and applied statistical problems. In this article, we study several properties of the truncated multivariate skew-normal distribution. We obtain distributional results through affine transformations, marginalization, and conditioning. Furthermore, the log-concavity of the joint probability density function is established. Full article
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17 pages, 554 KiB  
Article
Reliability Estimation for Stress-Strength Model Based on Unit-Half-Normal Distribution
Symmetry 2022, 14(4), 837; https://doi.org/10.3390/sym14040837 - 18 Apr 2022
Cited by 10 | Viewed by 2128
Abstract
Many lifetime distribution models have successfully served as population models for risk analysis and reliability mechanisms. We propose a novel estimation procedure of stress–strength reliability in the case of two independent unit-half-normal distributions can fit asymmetrical data with either positive or negative skew, [...] Read more.
Many lifetime distribution models have successfully served as population models for risk analysis and reliability mechanisms. We propose a novel estimation procedure of stress–strength reliability in the case of two independent unit-half-normal distributions can fit asymmetrical data with either positive or negative skew, with different shape parameters. We obtain the maximum likelihood estimator of the reliability, its asymptotic distribution, and exact and asymptotic confidence intervals. In addition, confidence intervals of model parameters are constructed by using bootstrap techniques. We study the performance of the estimators based on Monte Carlo simulations, the mean squared error, average bias and length, and coverage probabilities. Finally, we apply the proposed reliability model in data analysis of burr measurements on the iron sheets. Full article
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13 pages, 502 KiB  
Article
Asymmetric versus Symmetric Binary Regresion: A New Proposal with Applications
Symmetry 2022, 14(4), 733; https://doi.org/10.3390/sym14040733 - 04 Apr 2022
Cited by 2 | Viewed by 1750
Abstract
The classical logit and probit models allow to explain a dichotomous dependent variable as a function of factors or covariates which can influence the response variable. This paper introduces a new skew-logit link for item response theory by considering the arctan transformation over [...] Read more.
The classical logit and probit models allow to explain a dichotomous dependent variable as a function of factors or covariates which can influence the response variable. This paper introduces a new skew-logit link for item response theory by considering the arctan transformation over the scobit logit model, yielding a very flexible link function from a new class of generalized distribution. This approach assumes an asymmetric model, which reduces to the standard logit model for a special case of the parameters that control the distribution’s symmetry. The model proposed is simple and allows us to estimate the parameters without using Bayesian methods, which requires implementing Markov Chain Monte Carlo methods. Furthermore, no special function appears in the formulation of the model. We compared the proposed model with the classical logit specification using three datasets. The first one deals with the well-known data collection widely studied in the statistical literature, concerning with mortality of adult beetle after exposure to gaseous carbon disulphide, the second one considers an automobile insurance portfolio. Finally, the third dataset examines touristic data related to tourist expenditure. For these examples, the results illustrate that the new model changes the significance level of some explanatory variables and the marginal effects. For the latter example, we have also modified the definition of the intercept in the linear predictor to prevent confounding. Full article
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19 pages, 492 KiB  
Article
A Type I Generalized Logistic Distribution: Solving Its Estimation Problems with a Bayesian Approach and Numerical Applications Based on Simulated and Engineering Data
Symmetry 2022, 14(4), 655; https://doi.org/10.3390/sym14040655 - 24 Mar 2022
Cited by 3 | Viewed by 1810
Abstract
The family of logistic type distributions has been widely studied and applied in the literature. However, certain estimation problems exist in some members of this family. Particularly, the three-parameter type I generalized logistic distribution presents these problems, where the parameter space must be [...] Read more.
The family of logistic type distributions has been widely studied and applied in the literature. However, certain estimation problems exist in some members of this family. Particularly, the three-parameter type I generalized logistic distribution presents these problems, where the parameter space must be restricted for the existence of their maximum likelihood estimators. In this paper, motivated by the complexities that arise in the inference under the likelihood approach utilizing this distribution, we propose a Bayesian approach to solve these problems. A simulation study is carried out to assess the performance of some posterior distributional characteristics, such as the mean, using Monte Carlo Markov chain methods. To illustrate the potentiality of the Bayesian estimation in the three-parameter type I generalized logistic distribution, we apply the proposed method to real-world data related to the copper metallurgical engineering area. Full article
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15 pages, 1079 KiB  
Article
Slash Truncation Positive Normal Distribution and Its Estimation Based on the EM Algorithm
Symmetry 2021, 13(11), 2164; https://doi.org/10.3390/sym13112164 - 11 Nov 2021
Cited by 4 | Viewed by 1552
Abstract
In this paper, we present an extension of the truncated positive normal (TPN) distribution to model positive data with a high kurtosis. The new model is defined as the quotient between two random variables: the TPN distribution (numerator) and the power of a [...] Read more.
In this paper, we present an extension of the truncated positive normal (TPN) distribution to model positive data with a high kurtosis. The new model is defined as the quotient between two random variables: the TPN distribution (numerator) and the power of a standard uniform distribution (denominator). The resulting model has greater kurtosis than the TPN distribution. We studied some properties of the distribution, such as moments, asymmetry, and kurtosis. Parameter estimation is based on the moments method, and maximum likelihood estimation uses the expectation-maximization algorithm. We performed some simulation studies to assess the recovery parameters and illustrate the model with a real data application related to body weight. The computational implementation of this work was included in the tpn package of the R software. Full article
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21 pages, 6996 KiB  
Article
Exponentiated Generalized Inverted Gompertz Distribution: Properties and Estimation Methods with Applications to Symmetric and Asymmetric Data
Symmetry 2021, 13(10), 1868; https://doi.org/10.3390/sym13101868 - 04 Oct 2021
Cited by 2 | Viewed by 1423
Abstract
In this article, a new four-parameter lifetime model called the exponentiated generalized inverted Gompertz distribution is studied and proposed. The newly proposed distribution is able to model the lifetimes with upside-down bathtub-shaped hazard rates and is suitable for describing the negative and positive [...] Read more.
In this article, a new four-parameter lifetime model called the exponentiated generalized inverted Gompertz distribution is studied and proposed. The newly proposed distribution is able to model the lifetimes with upside-down bathtub-shaped hazard rates and is suitable for describing the negative and positive skewness. A detailed description of some various properties of this model, including the reliability function, hazard rate function, quantile function, and median, mode, moments, moment generating function, entropies, kurtosis, and skewness, mean waiting lifetime, and others are presented. The parameters of the studied model are appreciated using four various estimation methods, the maximum likelihood, least squares, weighted least squares, and Cramér-von Mises methods. A simulation study is carried out to examine the performance of the new model estimators based on the four estimation methods using the mean squared errors (MSEs) and the bias estimates. The flexibility of the proposed model is clarified by studying four different engineering applications to symmetric and asymmetric data, and it is found that this model is more flexible and works quite well for modeling these data. Full article
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18 pages, 657 KiB  
Article
A Class of Exponentiated Regression Model for Non Negative Censored Data with an Application to Antibody Response to Vaccine
Symmetry 2021, 13(8), 1419; https://doi.org/10.3390/sym13081419 - 03 Aug 2021
Cited by 1 | Viewed by 1377
Abstract
In this paper, an asymmetric regression model for censored non-negative data based on the centred exponentiated log-skew-normal and Bernoulli distributions mixture is introduced. To connect the discrete part with the continuous distribution, the logit link function is used. The parameters of the model [...] Read more.
In this paper, an asymmetric regression model for censored non-negative data based on the centred exponentiated log-skew-normal and Bernoulli distributions mixture is introduced. To connect the discrete part with the continuous distribution, the logit link function is used. The parameters of the model are estimated by using the likelihood maximum method. The score function and the information matrix are shown in detail. Antibody data from a study of the measles vaccine are used to illustrate applicability of the proposed model, and it was found the best fit to the data with respect to an others models used in the literature. Full article
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14 pages, 361 KiB  
Article
A Compound Class of the Inverse Gamma and Power Series Distributions
Symmetry 2021, 13(8), 1328; https://doi.org/10.3390/sym13081328 - 23 Jul 2021
Cited by 7 | Viewed by 1407
Abstract
In this paper, the inverse gamma power series (IGPS) class of distributions asymmetric is introduced. This family is obtained by compounding inverse gamma and power series distributions. We present the density, survival and hazard functions, moments and the order statistics of the IGPS. [...] Read more.
In this paper, the inverse gamma power series (IGPS) class of distributions asymmetric is introduced. This family is obtained by compounding inverse gamma and power series distributions. We present the density, survival and hazard functions, moments and the order statistics of the IGPS. Estimation is first discussed by means of the quantile method. Then, an EM algorithm is implemented to compute the maximum likelihood estimates of the parameters. Moreover, a simulation study is carried out to examine the effectiveness of these estimates. Finally, the performance of the new class is analyzed by means of two asymmetric real data sets. Full article
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16 pages, 993 KiB  
Article
The Skewed-Elliptical Log-Linear Birnbaum–Saunders Alpha-Power Model
Symmetry 2021, 13(7), 1297; https://doi.org/10.3390/sym13071297 - 19 Jul 2021
Cited by 3 | Viewed by 1376
Abstract
In this paper, the skew-elliptical sinh-alpha-power distribution is developed as a natural follow-up to the skew-elliptical log-linear Birnbaum–Saunders alpha-power distribution, previously studied in the literature. Special cases include the ordinary log-linear Birnbaum–Saunders and skewed log-linear Birnbaum–Saunders distributions. As shown, it is able to [...] Read more.
In this paper, the skew-elliptical sinh-alpha-power distribution is developed as a natural follow-up to the skew-elliptical log-linear Birnbaum–Saunders alpha-power distribution, previously studied in the literature. Special cases include the ordinary log-linear Birnbaum–Saunders and skewed log-linear Birnbaum–Saunders distributions. As shown, it is able to surpass the ordinary sinh-normal models when fitting data sets with high (above the expected with the sinh-normal) degrees of asymmetry. Maximum likelihood estimation is developed with the inverse of the observed information matrix used for standard error estimation. Large sample properties of the maximum likelihood estimators such as consistency and asymptotic normality are established. An application is reported for the data set previously analyzed in the literature, where performance of the new distribution is shown when compared with other proposed alternative models. Full article
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22 pages, 419 KiB  
Article
An Alternative One-Parameter Distribution for Bounded Data Modeling Generated from the Lambert Transformation
Symmetry 2021, 13(7), 1190; https://doi.org/10.3390/sym13071190 - 01 Jul 2021
Cited by 6 | Viewed by 1777
Abstract
The beta and Kumaraswamy distributions are two of the most widely used distributions for modeling bounded data. When the histogram of a certain dataset exhibits increasing or decreasing behavior, one-parameter distributions such as the power, Marshall–Olkin extended uniform and skew-uniform distributions become viable [...] Read more.
The beta and Kumaraswamy distributions are two of the most widely used distributions for modeling bounded data. When the histogram of a certain dataset exhibits increasing or decreasing behavior, one-parameter distributions such as the power, Marshall–Olkin extended uniform and skew-uniform distributions become viable alternatives. In this article, we propose a new one-parameter distribution for modeling bounded data, the Lambert-uniform distribution. The proposal can be considered as a natural alternative to well known one-parameter distributions in the statistical literature and, in certain scenarios, a viable alternative even for the two-parameter beta and Kumaraswamy distributions. We show that the density function of the proposal tends to positive finite values at the ends of the support, a behavior that favors good performance in certain scenarios. The raw moments are derived from the moment-generating function and used to describe the skewness and kurtosis behavior. The quantile function is expressed in closed form in terms of the Lambert W function, which allows reparameterizing the distribution such that the involved parameter represents the qth quantile. Thus, for the analysis of a bounded range variable, for which a set of covariates is available, we propose a regression model that relates the qth quantile of the response to a linear predictor through a link function. The parameter estimation is carried out using the maximum likelihood method and the behavior of the estimators is evaluated through simulation experiments. Finally, three application examples are considered in order to illustrate the usefulness of the proposal. Full article
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